Question

Wristwatch Accuracy Use the sample data from Exercise 7 “Wristwatch Accuracy” and construct a 95% confidence interval estimate of $\sigma$.

Short answer

The 95% confidence interval estimate for $\sigma$ is (184.0 sec, 441.1 sec).

Step-by-step solution

Given information

The data for the times (sec) for the discrepancy between the real time and the time indicated on the wristwatch is provided.

The level of confidence is 95%.

Compute the mean and standard deviation

Let x represents thetimes (sec) for the discrepancy between the real time and the time indicated on the wristwatch.

The sample mean is computed as,

$$\begin{gathered} \overline{x}=\frac{\sum x}{n} \\ =\frac{-85+325+20+305-93+...+36}{12} \\ =143 \end{gathered}$$

Therefore, the sample mean is 143.

The sample standard deviation is computed as,

$$\begin{gathered} s=\sqrt{\frac{\sum {\left(x-\overline{x}\right)}^2}{n-1}} \\ =\sqrt{\frac{{\left(-85-143\right)}^2+{\left(325-143\right)}^2+...+{\left(36-143\right)}^2}{12-1}} \\ =259.775 \end{gathered}$$

Therefore, the standard deviation is 259.775.

Compute the critical values

The level of confidence is 95%, which implies that the level of significance is 0.05.

The degrees of freedom is computed as,

$$\begin{gathered} \mathit{d}\mathit{f}\mathbf{=}\mathit{n}\mathbf{-}\mathbf{1} \\ =12-1 \\ =11 \end{gathered}$$

From the ${\chi }^2$ table, the critical values at 11 degrees of freedom and at 0.05 level of significance are 3.8157 and 21.92.

Construct the confidence interval estimate of  σ

The 95% confidence interval is given as,

$$\begin{gathered} \sqrt{\frac{\left(n-1\right){\mathbf{s}}^{\mathbf{2}}}{{\mathbf{\chi }}_{\mathbf{R}}^{\mathbf{2}}}}\mathbf{<}\mathit{\sigma }\mathbf{<}\sqrt{\frac{\left(n-1\right){\mathbf{s}}^{\mathbf{2}}}{{\mathbf{\chi }}_{\mathbf{L}}^{\mathbf{2}}}} \\ \sqrt{\frac{\left(12-1\right){259.775}^2}{21.92}}<\sigma <\sqrt{\frac{\left(12-1\right){259.775}^2}{3.8157}} \\ 184.0<\sigma <441.1 \end{gathered}$$

Therefore, the confidence interval estimate for $\sigma$ is $\left(184.0\sec ,441.1\sec \right)$.